Random high-dimensional orders

1995 ◽  
Vol 27 (1) ◽  
pp. 161-184 ◽  
Author(s):  
Béla Bollobás ◽  
Graham Brightwell
Keyword(s):  
Partial Order ◽  
Rapid Growth ◽  
Entire Range ◽  
Maximum Degree ◽  
Threshold Function ◽  
High Dimensional ◽  
Sharp Threshold ◽  
Threshold Functions

The random k-dimensional partial order Pk(n) on n points is defined by taking n points uniformly at random from [0,1]k. Previous work has concentrated on the case where k is constant: we consider the model where k increases with n.We pay particular attention to the height Hk(n) of Pk(n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in Pk(n): if k – (t/log t!) log n tends to + ∞ then the probability that Pk(n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of Hk(n) for the entire range of k(n).We also consider the maximum degree of Pk(n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k.We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of Pk(n) for large k(n).

Random high-dimensional orders

1995 ◽  
Vol 27 (01) ◽  
pp. 161-184 ◽  
Author(s):  
Béla Bollobás ◽  
Graham Brightwell
Keyword(s):  
Partial Order ◽  
Rapid Growth ◽  
Entire Range ◽  
Maximum Degree ◽  
Threshold Function ◽  
High Dimensional ◽  
Sharp Threshold ◽  
Threshold Functions

The random k-dimensional partial order P k (n) on n points is defined by taking n points uniformly at random from [0,1] k . Previous work has concentrated on the case where k is constant: we consider the model where k increases with n. We pay particular attention to the height H k (n) of P k (n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in P k (n): if k – (t/log t!) log n tends to + ∞ then the probability that P k (n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of H k (n) for the entire range of k(n). We also consider the maximum degree of P k (n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k. We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of P k (n) for large k(n).

Threshold Functions for H-factors

1993 ◽  
Vol 2 (2) ◽  
pp. 137-144 ◽  
Author(s):  
Noga Alon ◽  
Raphael Yuster
Keyword(s):  
Random Graph ◽  
Minimum Degree ◽  
Threshold Function ◽  
Sharp Threshold ◽  
Threshold Functions ◽  
Spanning Subgraph ◽  
Degree Of A Vertex ◽  
Vertex Disjoint ◽  
Special Case

Let H be a graph on h vertices, and G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n/h vertex disjoint copies of H. The fractional arboricity of H is , where the maximum is taken over all subgraphs (V′, E′) of H with |V′| > 1. Let δ(H) denote the minimum degree of a vertex of H. It is shown that if δ(H) < a(H), then n−1/a(H) is a sharp threshold function for the property that the random graph G(n, p) contains an H-factor. That is, there are two positive constants c and C so that for p(n) = cn−1/a(H) almost surely G(n, p(n)) does not have an H-factor, whereas for p(n) = Cn−1/a(H), almost surely G(n, p(n)) contains an H-factor (provided h divides n). A special case of this answers a problem of Erdős.

scholarly journals Sharp Threshold Functions for Random Intersection Graphs via a Coupling Method

10.37236/523 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Katarzyna Rybarczyk
Keyword(s):  
Perfect Matching ◽  
Hamilton Cycle ◽  
Coupling Method ◽  
New Method ◽  
Intersection Graphs ◽  
Sharp Threshold ◽  
Threshold Functions

We present a new method which enables us to find threshold functions for many properties in random intersection graphs. This method is used to establish sharp threshold functions in random intersection graphs for $k$–connectivity, perfect matching containment and Hamilton cycle containment.

STRUCTURAL CONNECTIONIST LEARNING WITH COMPLEMENTARY CODING

1992 ◽  
Vol 03 (01) ◽  
pp. 19-30 ◽  
Author(s):  
AKIRA NAMATAME ◽  
YOSHIAKI TSUKAMOTO
Keyword(s):  
Learning Algorithm ◽  
Internal Representation ◽  
Threshold Function ◽  
Sufficient Condition ◽  
Structural Learning ◽  
Similarity Matrix ◽  
Training Set ◽  
Threshold Functions ◽  
Connectionist Networks ◽  
Hidden Layer

We propose a new learning algorithm, structural learning with the complementary coding for concept learning problems. We introduce the new grouping measure that forms the similarity matrix over the training set and show this similarity matrix provides a sufficient condition for the linear separability of the set. Using the sufficient condition one should figure out a suitable composition of linearly separable threshold functions that classify exactly the set of labeled vectors. In the case of the nonlinear separability, the internal representation of connectionist networks, the number of the hidden units and value-space of these units, is pre-determined before learning based on the structure of the similarity matrix. A three-layer neural network is then constructed where each linearly separable threshold function is computed by a linear-threshold unit whose weights are determined by the one-shot learning algorithm that requires a single presentation of the training set. The structural learning algorithm proceeds to capture the connection weights so as to realize the pre-determined internal representation. The pre-structured internal representation, the activation value spaces at the hidden layer, defines intermediate-concepts. The target-concept is then learned as a combination of those intermediate-concepts. The ability to create the pre-structured internal representation based on the grouping measure distinguishes the structural learning from earlier methods such as backpropagation.

Depletion of retinal dopamine does not affect the ERG b-wave increment threshold function in goldfish in vivo

1994 ◽  
Vol 11 (4) ◽  
pp. 695-702 ◽  
Author(s):  
Zheng-Shi Lin ◽  
Stephen Yazulla
Keyword(s):  
Horizontal Cell ◽  
Threshold Function ◽  
Photopic Vision ◽  
Threshold Functions ◽  
Increment Threshold ◽  
Intensity Response ◽  
Purkinje Shift ◽  
6 Hydroxydopamine ◽  
Fish Retina

AbstractIncrement threshold functions of the electroretinogram (ERG) b–wave were obtained from goldfish using an in vivo preparation to study intraretinal mechanisms underlying the increase in perceived brightness induced by depletion of retinal dopamine by 6–hydroxydopamine (6–OHDA). Goldfish received unilateral intraocular injections of 6–OHDA plus pargyline on successive days. Depletion of retinal dopamine was confirmed by the absence of tyrosine-hydroxylase immunoreactivity at 2 to 3 weeks postinjection as compared to sham-injected eyes from the same fish. There was no difference among normal, sham-injected or 6–OHDA-injected eyes with regard to ERG waveform, intensity-response functions or increment threshold functions. Dopamine-depleted eyes showed a Purkinje shift, that is, a transition from rod-to-cone dominated vision with increasing levels of adaptation. We conclude (1) dopamine-depleted eyes are capable of photopic vision; and (2) the ERG b–wave is not diagnostic for luminosity coding at photopic backgrounds. We also predict that (1) dopamine is not required for the transition from scotopic to photopic vision in goldfish; (2) the ERG b–wave in goldfish is influenced by chromatic interactions; (3) horizontal cell spinules, though correlated with photopic mechanisms in the fish retina, are not necessary for the transition from scotopic to photopic vision; and (4) the OFF pathway, not the ON pathway, is involved in the action of dopamine on luminosity coding in the retina.

Improved Wavelet Thresholding for CCD Measurement Image Denoising

2014 ◽  
Vol 574 ◽  
pp. 432-435 ◽  
Author(s):  
Jie Zhan ◽  
Zhen Xing Li
Keyword(s):  
Image Denoising ◽  
Threshold Function ◽  
Wavelet Thresholding ◽  
Wavelet Coefficients ◽  
Local Correlation ◽  
Thresholding Method ◽  
Threshold Functions ◽  
Soft Threshold ◽  
Simulation Results ◽  
Filtering Effect

An improved wavelet thresholding method is presented and successfully applied to CCD measuring image denoising. On the analysis of the current widely used soft threshold and hard threshold, combining characteristics of the CCD measuring image and use of local correlation of wavelet coefficients, an improved threshold function is proposed, and the denoising results were contrasted among different threshold functions. The simulation results show that adopting the improved threshold function can acquire better filtering effect than traditional soft threshold and hard threshold methods.

Optimization of threshold functions over streams

2021 ◽  
Vol 14 (6) ◽  
pp. 878-889
Author(s):  
Walter Cai ◽  
Philip A. Bernstein ◽  
Wentao Wu ◽  
Badrish Chandramouli
Keyword(s):  
Data Stream ◽  
Broad Class ◽  
Threshold Function ◽  
Normal Operation ◽  
Time Interval ◽  
Surprising Result ◽  
Data Stream Management ◽  
Processing Application ◽  
Threshold Functions ◽  
Stream Management

A common stream processing application is alerting, where the data stream management system (DSMS) continuously evaluates a threshold function over incoming streams. If the threshold is crossed, the DSMS raises an alarm. The threshold function is often calculated over two or more streams, such as combining temperature and humidity readings to determine if moisture will form on a machine and therefore cause it to malfunction. This requires taking a temporal join across the input streams. We show that for the broad class of functions called quasiconvex functions, the DSMS needs to retain very few tuples per-data-stream for any given time interval and still never miss an alarm. This surprising result yields a large memory savings during normal operation. That savings is also important if one stream fails, since the DSMS would otherwise have to cache all tuples in other streams until the failed stream recovers. We prove our algorithm is optimal and provide experimental evidence that validates its substantial memory savings.

scholarly journals A New Wavelet Threshold Function and Denoising Application

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Lu Jing-yi ◽  
Lin Hong ◽  
Ye Dong ◽  
Zhang Yan-sheng
Keyword(s):  
Threshold Value ◽  
Threshold Function ◽  
Basic Principles ◽  
Threshold Functions ◽  
Fixed Threshold ◽  
Adjustment Factors ◽  
Soft Threshold ◽  
Traditional Structures ◽  
Mean Square Errors ◽  
Constant Deviation

In order to improve the effects of denoising, this paper introduces the basic principles of wavelet threshold denoising and traditional structures threshold functions. Meanwhile, it proposes wavelet threshold function and fixed threshold formula which are both improved here. First, this paper studies the problems existing in the traditional wavelet threshold functions and introduces the adjustment factors to construct the new threshold function basis on soft threshold function. Then, it studies the fixed threshold and introduces the logarithmic function of layer number of wavelet decomposition to design the new fixed threshold formula. Finally, this paper uses hard threshold, soft threshold, Garrote threshold, and improved threshold function to denoise different signals. And the paper also calculates signal-to-noise (SNR) and mean square errors (MSE) of the hard threshold functions, soft thresholding functions, Garrote threshold functions, and the improved threshold function after denoising. Theoretical analysis and experimental results showed that the proposed approach could improve soft threshold functions with constant deviation and hard threshold with discontinuous function problems. The proposed approach could improve the different decomposition scales that adopt the same threshold value to deal with the noise problems, also effectively filter the noise in the signals, and improve the SNR and reduce the MSE of output signals.

Application of an Improved Wavelet Threshold Denoising Method for Vibration Signal Processing

2014 ◽  
Vol 889-890 ◽  
pp. 799-806 ◽  
Author(s):  
Zhi Jie Xie ◽  
Bao Yu Song ◽  
Yang Zhang ◽  
Feng Zhang
Keyword(s):  
Signal Processing ◽  
Wavelet Decomposition ◽  
Wavelet Coefficient ◽  
Mechanical Vibration ◽  
Vibration Signal ◽  
Threshold Function ◽  
Denoising Method ◽  
Threshold Functions ◽  
Vibration Signal Analysis ◽  
White Noises

Vibration signal analysis has been widely used in the fault detection and condition monitoring of rotation machinery. But the practical signals are easily polluted by noises in their transmission process. The raw signals should be processed to reduce noise and improve the quality before further analyzing. In this paper an improved wavelet threshold denosing method for vibration signal processing is proposed. Firstly, a new threshold is developed based on the VisuShrink threshold. The effect of noise standard deviation and wavelet coefficient is retained, and the correlation of wavelet decomposition scale is considered. Then, a new threshold function is defined. The new algorithm is able to overcome the discontinuity in hard threshold denoising method and reduce the distortion caused by permanent bias of wavelet coefficient in soft threshold denoising method. At last five kinds of threshold principles and three kinds of threshold functions are compared in processing the same signal, which is simulated as the mechanical vibration signal added white noises. The results show that the improved threshold is superior to the traditional threshold principles and the new threshold function is more effective than soft and hard threshold function in improving SNR and decreasing RMSE.

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